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arXiv · 2601.14064

Time-dependent metrics and connections

Xavier Gràcia·Xavier Rivas·Daniel Torres
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Abstract

Time-dependent structures often appear in differential geometry, particularly in the study of non-autonomous differential equations on manifolds. One may study the geodesics associated with a time-dependent Riemannian metric by extremizing the corresponding energy functional, but also through the introduction of a more general concept of time-dependent covariant derivative operator. This relies on the examination of connections on the product manifold $\mathbb{R}\times M$. For these time-dependent covariant derivatives we explore the notions of parallel transport, geodesics and torsion. We also define the derivative of a one-parameter family of connections.

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Xavier Gràcia, Xavier Rivas, Daniel Torres. 2026-01-20. Time-dependent metrics and connections. https://arxiv.org/abs/2601.14064

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